TRIANGLE. 



Wherefore, if in two triangles A C B mdaci {PfateXV. 

 Geometry, fg. g.) A C : A B :: a c : a b ; A C : C B :: a c 

 : i c ; the triangles are determined in the fame manner, and 

 confequently are fimilar, and therefore mutually equiangular. 



3. A right line, as A B, and two adjacent angles A and 

 B, which, taken together, are lefs than two right ones, 

 being given ; to defcribe the triangle ABC. On the 

 given line A B, make the two given angles A and B : con- 

 tinue the fides A C and B C, till they meet in C. Then 

 will A B C be the triangle required. 



Hence, one fide and two angles beisg given, the whole 

 triangle is determined. Wherefore, if in two triangles 

 A ■=■ a and B = ^, the triangles are determined after the 

 fame manner, and therefore are fimilar. 



Triangles, Menfuration of. To find the area of a tri- 

 angle, multiply the bafe A B {Jig. 10.) by the altitude- 

 C «/; half the produft is the area of the triangle ABC. 



Or thus ; multiply half the bafe A B by the altitude 

 Q d ; or the whole bafe by half the altitude ; the produft is 

 the area of the triangle. 



•,?''• 



See Quadrature. 



Or, the area of any triangle is had by adding all the 

 three fides together, and taking half the fum ; and from 

 that half fum fubtrading each fide feverally, and multiply- 

 ing that half fum and the remainder continually into one 

 another, and extraftingthe fquare root of the produft. 



Hence, l. If between the bafe and half the altitude, 

 or between the altitude and half the bafe, be found a 

 mean proportional ; it will be the fide of a fquare equal 

 to the triangle. 2. If the area of a triangle be divided by 

 half the bafe, the quotient is the altitude. 



Triangles, Properties of Plane, i. If in two triangles 

 A B C and abc (fg. 9. ) the angle A be = a ; and the 

 fides A B = fl i, and A C = a c ; then will the fide B C = 

 b c, and the angles C = c, and B = i ; and therefore the 

 ■whole triangles will be equal and fimilar. 



2. If one fide of a triangle ABC [fg. n.) be con- 

 tinued to D, the external angle DAB will be greater 

 than either of the internal oppofite ones B or C. 



3. In every triangle, the greateft fide is oppofed to the 

 greateft angle, and the leail to the leaft. 



4. In every triangle, any two fides taken together are 

 greater than the third. 



5. In two triangles, if the feveral fides of the one be 

 refpeftively equal to the fides of the other, the angles will 

 likewife be refpeftively equal ; and confequently the whole 

 triangles will be equal and fimilar. 



6. If any fide, as B C [fg. 12.) of a triangle AC B 

 be continued to D, the external angle DO A will be 

 equal to the two internal oppofite ones y and 2 taken 

 together. 



7. In every triangle, as ABC,- the three angles A, 

 B, C, taken together, are equal to two right ones, or 

 180'='. 



Hence, i . If the triangle be reAangular, as M K L 

 {Jig. 7.) the two oblique angles M and L, taken together, 

 make a right .angle, or 90° ; and therefore are half right, 



12 



if the triangle be ifofceles. 2. If one angle of a triangle 

 be oblique, the other two taken together are oblique like- 

 wife. 3. In an equilateral triangle, each angle is 60*^. 



4. If one angle of a triangle be fubtrafted from 1 80°, the 

 remainder is the fum of the other two ; and if the fum of 

 two be fubtrafted from 180°, the remainder is the third. 



5. If two angles of one triangle be equal to two of an- 

 other, either together or feparately, the third of the one 

 muft be likewife equal to the third of the other. 6. Since 

 in an ifofceles triangle D F E {Plate VIII. fg. 105.) the 

 angles at the bafe y and a are equal ; if the angle at the 

 vertex be fubtrafted from 180°, and the remainder be di- 

 vided by 2, the quotient is the quantity of each of the 

 equal angles : in hke manner, if the double of one of the 

 angles at the bafe y be fubtrafted from 1 80°, the remainder 

 is the quantity of the angle at the vertex. See IsosCELES 

 Triangle. 



8. If in two triangles ABC and abc {Plate XV. 

 Geometry, fg. 9.) A3 =: a b, A = a, and B = i ; then 

 will AC = a f, B C = i c, C = f, and the triangle A C B 

 equal and fimilar to the triangle abc. Hence, if in two 

 triangles A C B and acb, A = a, B = ^, and B C =^ bci 

 then will C = c ; confequently AC := a c, AB =ai; 

 and the triangle A C 3 = a c b. 



9. If in a triangle D F E, the angles at the bafe y and « 

 {Plate Ylll. Geometry, fg. 105.) be equal, the triangle is 

 ifofceles : confequently, if the three angles be equal, it is 

 equilateral. 



10. If in a triangle A B C ( Plate XV. fg. 14. ) a right 

 line D E be drawn parallel to the bafe A C, then will B A : 

 BC:^BD:BE::AD:EC; andBA:AC:: 

 B D : D E ; confequently the triangle B D E fimilar to 

 B A C. And, vice verfa, a right line, which divides two 

 fides of a triangle proportionally, is parallel to the remain- 

 ing fide. Moreover, if another right line F G be alfo drawn 

 parallel to the bafe A C, the intercepted parts, D F, EG, 

 are in the fame ratio with the whole fides A B, C B ; i. e. 

 DF:EG::AB:BC. And if any number of lines be 

 drawn parallel to the bafe, cutting the fides of a triangle, 

 evei-y two correfponding fegments will have the fame ratio. 



1 1. Every triangle may be infcribed in a circle. 



12. The fide of .an equilateral hexagon, infcribed in a 

 circle, is equal to the radius. 



13. Triangles on the fame bafe, and having the fame 

 height, that is, being between tlie fame parallel lines, are 

 always equal. See Parallelogram. 



14. Every triangk, as CAD, {Plate X. Geometry, 

 _/ff. 14.) is one-half of a parallelogram A C D B on the 



fame, or an equal bafe CD, and of the fame altitude, 

 or between the fame parallels : or a triangle is equal to 

 a parallelogram upon the fame bafe> but half the ahi- 

 tude ; or half the bafe, and the fame altitude. See Pa- 

 rallelogram. 



1 5. In every triangle, as well plane as fpherical, the fides 

 or fines of the fides are proportional to the fines of the op- 

 pofite angles. 



16. In every plane triangle, as the fum of two fides is to 

 their difference, fo is the tangent of half the fum of the op- 

 pofite angles, to the tangent of half their difference. See 

 Tangent. 



17. If a perpendicular be let fall upon the bafe of an ob- 

 lique-angled triangle, the difference of the fquares, or, the 

 reftangle under the fum and ditTercnce, of the fides is equal 

 to double the reftangle under the bafe, and the diftance ot 

 the perpendicular from the middle of the bale. 



18. The double of the fquare of a line drawn from ttie 

 vertex to the middle of the bafe of any triangle, togetfier 



with 



